Vector spaces and subspaces,linearly dependent and independent sets of vectors, space spanned by a set of vectors, basis of a vector space, sum and intersection space of two vector spaces, coordinate systems in vector spaces, changes in coordinates due to change in basis

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Vector spaces and subspaces, linearly dependent and independent sets of vectors, space spanned by a set of vectors, basis of a vector space, sum and intersection space of two vector spaces, coordinate systems in vector spaces, changes in coordinates due to change in basis

Def. Vector space. Any set of n-vectors over a field F which is closed under both addition and scalar multiplication is called a vector space.

Closure under addition. A set of vectors is said to be closed under addition if the sum of any two of them is a vector of the set.

Closure under scalar multiplication. A set of vectors is said to be closed under scalar multiplication if every scalar multiple of a vector of the set is a vector of the set.

Example 1. Let Q be the set of all real 2-tuples (x₁, x₂). This set Q is closed with respect to both vector addition and scalar multiplication: for any two scalars a and b and any two vectors X and Y, the vector Z = aX + bY is a member of Q. Thus Q is a vector space. It is Real 2-space, V₂(R) i.e. it is the set of all vectors extending from the origin to all points of the plane.

Example 2. Let Q be the set of all real 3-tuples (x₁, x₂, x₃). This set Q is closed with respect to both vector addition and scalar multiplication: for any two scalars a and b and any two vectors X and Y, the vector Z = aX + bY is a member of Q. Thus Q is a vector space. It is Real 3-space, V₃(R) i.e. it is the set of all vectors extending from the origin to all points of three dimensional space.

Example 3. Let Q be the set of all real 4-tuples (x₁, x₂, x₃, x₄). This set Q is closed with respect to both vector addition and scalar multiplication: for any two scalars a and b and any two vectors X and Y, the vector Z = aX + bY is a member of Q. Thus Q is a vector space. It is Real 4-space, V₄(R).

Example 4. Let X₁ = (2, 5, 4), and X₂ = (9, 3, 8), two vectors in Real 3-space, V₃(R). Then

Z = k₁ X₁ + k₁ X_{2 ,}

where k₁, k₂, ... ,k_m are scalars, is a vector space. It is a two dimensional subspace of V₃(R). It corresponds to that plane passing through the two vectors X₁ and X_2.

Linear combination of vectors. The vector c₁x₁ + c₂x₂ + ... + c_mx_m with arbitrary numerical values for the coefficients c₁, c₂, ... ,c_m is called a linear combination of the vectors x₁, x₂, ... ,x_m .

Theorem. If X₁, X₂, ... ,X_m are m n-vectors of V_n(F), the set of all linear combinations

k₁X₁ + k₂X₂ + ... + k_mX_m

where k₁, k₂, ... ,k_m are scalars, is a vector space.

If, in the above theorem, m < n, the space corresponding to the product

k₁X₁ + k₂X₂ + ... + k_mX_m

will be a subspace of the vector space V_n(F). As an example, see Example 4 above.

Vector spaces and subspaces – examples.

1. Let A and B be any two non-collinear vectors in the x-y plane. Then any other vector X in the plane can be expressed as a linear combination of vectors A and B. That is there exist numbers k₁ and k₂ such that X = k₁A + k₂B for any vector X. Conversely, any linear combination of vectors A and B gives a vector in the x-y plane. Note the closure idea involved. Any vector in the plane can be obtained as a linear combination of A and B and any linear combination gives some vector in the plane. It is a closed system. It is a vector space.

2. Let A, B and C be any three non-coplanar vectors in an x-y-z Cartesian coordinate system. Then any vector in this x-y-z coordinate system can be expressed as a linear combination of A, B and C. That is, there exist numbers k₁, k₂, and k₃ such that X = k₁A + k₂B + k₃C for any vector X. Conversely, any linear combination of A, B and C gives some vector in the x-y-z coordinate system. Note the closure idea involved. It is a closed system. It is a vector space. Let us call it “Vector space Q”.

3. Pass any plane through the origin of an x-y-z Cartesian coordinate system. Denote the plane by K. Let A and B be any two non-collinear vectors lying in plane K. Then any linear combination of vectors A and B is a vector lying in plane K (i.e. if c₁ and c₂ are any two numbers, the vector X = c₁A + c₂B lies in plane K). Moreover, any vector lying in plane K can be expressed as a linear combination of vectors A and B (i.e. for any vector X in plane K there exist numbers c₁ and c₂ such that X = c₁A + c₂B ). Note the closure concept involved. It is a closed system. The totality of all vectors in plane K constitute a vector space. This space represents a two-dimensional subspace of Vector space Q above.

4. Pass any line through the origin of an x-y-z Cartesian coordinate system. Denote the line by L. Let A be any vector lying in the line. Then any multiple of vector A is a vector lying in line L. Moreover, any vector lying in line L can be expressed as a multiple of vector A. Note the closure concept involved. It is a closed system. The totality of all vectors in line L constitute a vector space. This space represents a one-dimensional subspace of Vector space Q above.

Linearly dependent and independent sets of vectors. A set of vectors x₁, x₂, ... ,x_m is said to be linearly dependent if some one of the vectors in the set can be expressed as a linear combination of one or more of the other vectors in the set. If none of the vectors in the set can be expressed as a linear combination of any other vectors of the set, then the set is said to be linearly independent.

Examples from three-dimensional space. To illustrate the concepts let us consider some examples from three dimensional space.

1. Let A, B and C be any three non-coplanar vectors in an x-y-z Cartesian coordinate system. Then any vector in this x-y-z coordinate system can be expressed as a linear combination of A, B and C. However, none of these three vectors A, B and C can be expressed as a linear combination of the other two. The vectors A, B and C constitute a linearly independent set. Now add another vector D to the set. Consider the set A, B, C and D. This set is a dependent set because vector D can be expressed as a linear combination of A, B and C.

2. Pass any plane through the origin of an x-y-z Cartesian coordinate system. Denote the plane by K. Let A and B be any two non-collinear vectors lying in plane K. Then any linear combination of vectors A and B is a vector lying in plane K. However, neither vector A or B can be expressed as a linear combination of the other. The two vectors form a linearly independent set. Now add another vector C, also lying in plane K, to the set. The set A, B and C is a dependent set because vector C can be expressed as a linear combination of A and B.

A necessary and sufficient condition for the linear independence of a set of vectors. There exists an important algebraic criterion, an algebraic test, which can tell us whether a set of vectors is linearly independent or not. That test is given by the following theorem:

Theorem. A necessary and sufficient condition for the set of vectors x₁, x₂, ... , x_mto be linearly independent is that

c₁x₁ + c₂x₂ + ... + c_mx_m = 0

only when all the scalars c_i are zero.

What is the reasoning that leads to the assertion of this theorem? Well, a set of vectors x₁, x₂, ... ,x_m is linearly dependent if some one of the vectors in the set can be expressed as a linear combination of one or more of the other vectors in the set, that is if there exists some vector x_i in the set such that

x_i = a₁x_j + a₂x_k + ...

for one or more vectors x_j, x_k, etc. of the set. This condition implies that there exists some subset of vectors x_i, x_j, x_k, etc. within the full set such that

c_ix_i + c_jx_j + c_kx_k + ... = 0

where c_i c_j, c_k, etc. are non-zero. Said differently, a set is linearly dependent if there exist two or more non-zero c’s for which the following equation holds true:

c₁x₁ + c₂x₂ + ... + c_mx_m = 0

(i.e. it is possible for the equation to hold true even though not all of the c’s are zero). If there does not exist two or more non-zero c’s for which it will hold, then the set of vectors is linearly independent. The case in which only one of the c’s is non-zero is impossible since c_ix_i = 0 is not possible if c ≠ 0. Thus the set of vectors is linearly independent if and only if

c₁x₁ + c₂x₂ + ... + c_mx_m = 0

only when all the scalars c_i are zero.

Linear dependence or independence of a set of vectors is determined from the rank of a matrix formed from them. Consider a matrix formed from m n-vectors with each vector corresponding to a row in the matrix. If the rank of the matrix is m the set of vectors is linearly independent. If the rank is less than m the set of vectors is linearly dependent. If the rank r is less than m then there are exactly r vectors in the set which are linearly independent and the remaining vectors c₁x₁ + c₂x₂ + ... + c_mx_m can be expressed as a linear combination of these r independent vectors.

Space spanned by a set of vectors. Let x₁, x₂, ... ,x_m be a set of n-vectors in n-dimensional space. They may be a linearly independent set or a linearly dependent set of vectors, it doesn’t matter. The set of all linear combinations of these vectors corresponds either to all of n-dimensional space or to some subspace of n-dimensional space. The vector space generated by all linear combinations of x₁, x₂, ... ,x_m is called the subspace spanned by x₁, x₂, ... ,x_m.

Example. Let us illustrate the concept with an example from three-dimensional space.

Pass any plane through the origin of an x-y-z Cartesian coordinate system. Denote the plane by K. Let A, B, C, D and E be five non-collinear vectors lying in plane K. These five vectors form a set that spans plane K, a subspace of three dimensional space. Any linear combination of these vectors lies in plane K and no linear combination lies outside the plane. Now add to this set a vector F which lies outside this plane. The new set of vectors (vectors A, B, C, D, E and F) span all of three-dimensional space, Why? Because the set now contains three linearly independent vectors and three independent vectors will span all of three-dimensional space.

Basis of a vector space. A basis of a vector space is any set of linearly independent vectors that spans the space. Each vector of the space is then a unique linear combination of the vectors of this basis.

Examples.

1. In three-dimensional space any set of three non-coplanar vectors constitute a basis for the space (choose any three non-coplanar vectors and they qualify as a basis). Any vector in the space can be expressed as a linear combination of these basis vectors and, conversely, any linear combination of these three basis vectors lies in three dimensional space. This basis plays a role analogous to a set of coordinate axes (it is viewed as a non-orthogonal set of coordinate axes that serve as a reference frame from which we can express any other vector in the space).

2. Any two non-collinear vectors in three-dimensional space define a plane that constitutes a subspace of three-dimensional space since any linear combination of these two vectors lies in the plane and, conversely, any vector in the plane can be expressed as a linear combination of these two vectors. Thus these two vectors constitute a basis for a two-dimensional subspace of three-dimensional space. Similarly, a single vector in 3-space constitutes a basis for a one-dimensional subspace of 3-space.

3. In two-dimensional space any set of two non-collinear vectors constitute a basis for the space. These two basis vectors than serve as a non-orthogonal reference frame from which any other vector in the space can be expressed.

Dimension of a vector space. The dimension of a vector space is the number of independent vectors required to span the space.

Subspace notation. N-dimensional space V_n(F) has embedded in it subspaces of lesser dimensions. For example, ordinary three-dimensional space has embedded in it two-dimensional subspaces in the form of planes passing through the origin of the coordinate system and one-dimensional subspaces in the form of lines passing through the origin. A r-dimensional subspace of V_n(F) is denoted by V_n^r(F). A two-dimensional subspace of ordinary three-dimensional space V₃(R) would, for example, be denoted by V₃²(R).

Sum space of two vector spaces. The sum of two vector spaces P and Q is defined as the totality of all vectors x + y where x is in P and y is in Q. This is a vector space and we call it the sum space of P and Q. The sum space can be regarded as the space spanned by the union of the bases of the spaces P and Q.

Intersection space of two spaces. The intersection space of two vector spaces is the set of all vectors that belong to both spaces.

Example. A plane passing through the origin of an x-y-z Cartesian system in ordinary three-dimensional space represents a two-dimensional subspace of three-dimensional space. Consider two planes P and Q passing through the origin which are assumed not to coincide. The sum space of the two planes is the whole three dimensional space and the intersection space is a straight line (the line of their intersection).

Theorem. The dimensions p and q of two given spaces, the dimensions t of their sum and the dimension s of their intersection satisfy the following relation:

p + q = t + s

Coordinate systems in vector spaces. Consider a n-dimensional vector space with basis vectors A₁, A₂ , ... ,A_n so that an arbitrary vector X of the space has a unique representation

X = u₁A₁ + u₂A₂ + ... + u_nA_n

The vectors A₁, A₂ , ... ,A_n are called a coordinate system or reference system in the space and u₁, u₂, ... , u_n are the coordinates of X with respect to this system. Thus we can call the n-tuple {u₁, u₂, ... , u_n } the coordinate vector of X relative to the basis {A₁, A₂ , ... , A_n } and denote X by

where A denotes the basis {A₁, A₂ , ... , A_n }.

Elementary basis for n-space. The following set of vectors is a special set of vectors called the elementary basis (E-basis) for n-space [either Real n-space, V_n(R) or Complex n-space, V_n(C) ]:

E₁ = [1, 0, 0, ...., 0]^T

E₂ = [0, 1, 0, ...., 0]^T

..............................

E_n = [0, 0, 0, ...., 1]^T .

E-basis Coordinate System. The n-vectors

E₁ = [1, 0, 0, ...., 0]^T

E₂ = [0, 1, 0, ...., 0]^T

..............................

E_n = [0, 0, 0, ...., 1]^T

are called the elementary n-vectors, unit n-vectors or elementary unit vectors. The elementary n-vector E_j, whose j-th component is 1, is called the j-th elementary n-vector. The elementary n-vectors E₁, E₂, ... ,E_n constitute an important basis for V_n(F). Every vector X = [x₁, x₂, ... ,x_n]^T of n-space V_n(F), can be expressed uniquely as the sum

X = x₁E₁ + x₂E₂ + ... + x_nE_n

of the elementary vectors. The components x₁, x₂, ... ,x_n of X are now called the coordinates of X relative to the E-basis.

Changes in coordinates due to change in basis. Let us consider the problem of how the coordinates of vectors are changed on transition from one basis to another in an

n-dimensional vector space.

Let the original basis be the usual E-basis E₁, E₂, ... ,E_n . Let x₁, x₂, ... ,x_n be the coordinates of a vector X with respect to the E-basis . Let Z₁, Z₂, ... ,Z_n some arbitrary other basis. Then there exist unique numbers a₁, a₂, ... ,a_n such that

X = a₁Z₁ + a₂Z₂ + ... + a_nZ_n

These numbers a₁, a₂, ... ,a_n represent the coordinates of X relative to the Z-basis. Writing

we have X = [ Z₁, Z₂, ... ,Z_n ] X_Z = ZX_Z

where Z is the matrix [ Z₁, Z₂, ... ,Z_n ] whose columns are the basis vectors Z₁, Z₂, ... ,Z_n .

Thus we have the following result: the coordinates of a vector X with respect to the E-basis are related to its coordinates with respect to some other Z-basis by

X = ZX_Z

where matrix Z , whose columns are the new basis vectors Z₁, Z₂, ... ,Z_n , is called the “matrix of the coordinate transformation”.

Now let us ask what happens to the coordinates of a point if we move from some Z-basis given by {Z₁, Z₂, ... ,Z_n } to some other W-basis given by {W₁, W₂, ... ,W_n }. We know

X = ZX_Z

X = WX_W

WX_W = ZX_Z

and

X_W = W^-1ZX_Z

Theorem. If a vector of V_n(F) has coordinates X_Zand X_W relative to bases {Z₁, Z₂, ... ,Z_n } and {W₁, W₂, ... ,W_n } of V_n(F), there exits a nonsingular matrix P, determined solely by the two bases and given by P = W^-1Z, such that X_W = PX_Z .

References.

Ayres. Matrices (Schaum).